Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations

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Boltzmann equation and large deviations

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

To cite this version:

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella. Statistical dynamics

of a hard sphere gas: fluctuating Boltzmann equation and large deviations. 2020. �hal-02920308�

Thierry Bodineau Isabelle Gallagher Laure Saint-Raymond Sergio Simonella

STATISTICAL DYNAMICS OF A HARD SPHERE GAS:

FLUCTUATING BOLTZMANN EQUATION

AND LARGE DEVIATIONS

CMAP, CNRS, Ecole Polytechnique, I.P. Paris, Route de Saclay, 91128 Palaiseau Cedex, FRANCE.

E-mail :

thierry.bodineau@polytechnique.edu

I. Gallagher

DMA, ´ Ecole normale sup´erieure, CNRS, PSL Research University, 45 rue d’Ulm, 75005 Paris, FRANCE, and Universit´e de Paris.

E-mail :

gallagher@math.ens.fr

L. Saint-Raymond

UMPA UMR 5669 du CNRS, ENS de Lyon,Universit´e de Lyon, 46 all´ee d’Italie, 69007 Lyon, FRANCE.

E-mail :

Laure.Saint-Raymond@ens-lyon.fr

S. Simonella

UMPA UMR 5669 du CNRS, ENS de Lyon, Universit´e de Lyon, 46 all´ee d’Italie, 69007 Lyon, FRANCE.

E-mail :

sergio.simonella@ens-lyon.fr

We are very grateful to H. Spohn and M. Pulvirenti for many enlightening discussions on the subjects

treated in this text. We thank also F. Bouchet, F. Rezakhanlou, G. Basile, D. Benedetto, L. Bertini

for sharing their insights on large deviations and A. Debussche, A. de Bouard, J. Vovelle for their

explanations on SPDEs.

STATISTICAL DYNAMICS OF A HARD SPHERE GAS:

FLUCTUATING BOLTZMANN EQUATION AND LARGE DEVIATIONS

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

Abstract. —

We present a mathematical theory of dynamical fluctuations for the hard sphere gas in the Boltzmann-Grad limit. We prove that: (1) fluctuations of the empirical measure from the solution of the Boltzmann equation, scaled with the square root of the average number of particles, converge to a Gaussian process driven by the fluctuating Boltzmann equation, as predicted in [42]; (2) large deviations are exponentially small in the average number of particles and are characterized, under regularity assumptions, by a large deviation functional as previously obtained in [38] in a context of stochastic processes. The results are valid away from thermal equilibrium, but only for short times.

Our strategy is based on uniform a priori bounds on the cumulant generating function, characterizing

the fine structure of the small correlations.

CONTENTS

1. Introduction. . . . 1

1.1. The hard-sphere model. . . . 1

1.2. Lanford’s theorem : a law of large numbers. . . . 4

1.3. The fluctuating Boltzmann equation. . . . 5

1.4. Large deviations. . . . 7

1.5. Strategy of the proofs. . . . 8

1.6. Remarks, and open problems. . . 11

Part I. Dynamical cumulants . . . 13

2. Combinatorics on connected clusters. . . 15

2.1. Generating functionals and cumulants. . . 15

2.2. Inversion formula for cumulants. . . 17

2.3. Clusters and the tree inequality. . . 18

2.4. Number of minimally connected graphs. . . 20

2.5. Combinatorial identities. . . 21

3. Tree expansions of the hard-sphere dynamics. . . 23

3.1. Space correlation functions. . . 23

3.2. Geometrical representation with collision trees. . . 24

3.3. Averaging over trajectories. . . 25

4. Cumulants for the hard-sphere dynamics. . . 29

4.1. External recollisions. . . 29

4.2. Overlaps. . . 31

4.3. Initial clusters. . . 32

4.4. Dynamical cumulants. . . 33

5. Characterization of the limiting cumulants. . . 37

5.1. Limiting pseudo-trajectories and graphical representation of limiting cumulants. . . 37

5.2. Limiting cumulant generating function. . . 39

5.3. Hamilton-Jacobi equations. . . 41

5.4. Stability of the Hamilton-Jacobi equation. . . 46

5.5. Dynamical equations for the limiting cumulant densities. . . 48

Part II. Fluctuations around the Boltzmann dynamics . . . 51

6. Fluctuating Boltzmann equation. . . 53

6.1. Weak solutions for the limit process. . . 53

6.2. Convergence of the characteristic function. . . 57

6.3. Tightness and proof of Theorem 2. . . 59

6.4. The modified Garsia, Rodemich, Rumsey inequality. . . 64

6.5. Spohn’s formula for the covariance. . . 66

7. Large deviations. . . 69

7.1. Large deviation asymptotics. . . 69

7.2. Identification of the large deviation functionals F = F

b

. . . 74

Part III. Uniform a priori bounds and convergence of the cumulants . . . 83

8. Clustering constraints and cumulant estimates . . . 85

8.1. Dynamical constraints. . . 86

8.2. Decay estimate for the cumulants. . . 95

9. Minimal trees and convergence of the cumulants. . . 101

9.1. Truncated cumulants. . . 101

9.2. Removing non clustering recollisions/overlaps and non regular overlaps. . . 102

9.3. Proof of Theorem 10 : convergence of the cumulants. . . 107

9.4. Analysis of the geometric conditions. . . 108

CONTENTS 7

Bibliography. . . 115

Notation Index. . . 119

CHAPTER 1

INTRODUCTION

This paper is devoted to a detailed analysis of the correlations arising, at low density, in a deterministic particle system obeying Newton’s laws. In this chapter we start by defining our model precisely, and recalling the fundamental result of Lanford on the short-time validity of the Boltzmann equation. After that, we state our main results, Theorem 2 and Theorem 3 below, regarding small fluctuations and large deviations of the empirical measure, respectively. Finally, the last section of this introduction describes the essential features of the proof, the organization of the paper, and presents some open problems.

1.1. The hard-sphere model

We consider a system of N ≥ 0 spheres of diameter ε > 0 in the d-dimensional torus

T^dN

with d ≥ 2.

The positions (x

^ε₁

, . . . , x

^ε_N

) ∈

T^dN

and velocities (v

^ε₁

, . . . , v

^ε_N

) ∈

R^dN

of the particles satisfy Newton’s laws

(1.1.1) dx

^ε_i

dt = v

^ε_i

, dv

^ε_i

dt = 0 as long as | x

^ε_i

(t) − x

^ε_j

(t) | > ε for 1 ≤ i 6 = j ≤ N , with specular reflection at collisions

(1.1.2)

(v

^εi

)

⁰

:= v

_i^ε

− 1

ε

²

(v

i^ε

− v

_j^ε

) · (x

^εi

− x

^ε_j

) (x

^εi

− x

^ε_j

) v

^ε_j0

:= v

_j^ε

+ 1

ε

²

(v

i^ε

− v

_j^ε

) · (x

^εi

− x

^ε_j

) (x

^εi

− x

^ε_j

)







if | x

^ε_i

(t) − x

^ε_j

(t) | = ε .

Observe that these boundary conditions do not cover all possible situations, as for instance triple collisions are excluded. Nevertheless the hard-sphere flow generated by (1.1.1)-(1.1.2) (free transport of N spheres of diameter ε, plus instantaneous reflection

v

^ε_i

, v

_j^ε

→ v

^ε_i⁰

, v

^ε_j⁰

at contact) is well defined on a full measure subset of D

N^ε

(see [1], or [17] for instance) where D

N^ε

is the canonical phase space

D

N^ε

:=

Z

N

∈

D^N

/ ∀ i 6 = j , | x

i

− x

j

| > ε .

We have denoted Z

_N

:= (X

N

, V

_N

) ∈ (

T^d

×R

^d

)

^N

the positions and velocities in the phase space

D^N

:=

(T

^d

×

R^d

)

^N

with X

N

:= (x

1

, . . . , x

N

) ∈

T^dN

and V

N

:= (v

1

, . . . , v

N

) ∈

R^dN

. We set Z

N

= (z

1

, . . . , z

N

)

with z

i

= (x

i

, v

i

).

The probability density W

_N^ε

of finding N hard spheres of diameter ε at configuration Z

N

at time t is governed by the Liouville equation in the 2dN-dimensional phase space

(1.1.3) ∂

_t

W

_N^ε

+ V

_N

· ∇

^XN

W

_N^ε

= 0 on D

N^ε

, with specular reflection on the boundary. If we denote

∂ D

^ε±N

(i, j) :=

n

Z

N

∈

D^N

/ | x

i

− x

j

| = ε , ± (v

i

− v

j

) · (x

i

− x

j

) > 0

and ∀ k, ` ∈ [1 , N ]

²

\ { i, j } , k 6 = ` , | x

_k

− x

_`

| > ε

o

, then

(1.1.4) ∀ Z

N

∈ ∂ D

^ε+N

(i, j) , i 6 = j , W

_N^ε

(t, Z

N

) := W

_N^ε

(t, Z

_N^0i,j

) , where Z

_N^0i,j

differs from Z

N

only by (v

i

, v

j

) → v

⁰_i

, v

_j⁰

, given by (1.1.2).

The canonical formalism consists in fixing the number N of particles, and in studying the probability density W

_N^ε

of particles in the state Z

N

at time t, as well as its marginals. The main drawback of this formalism is that fixing the number of particles creates spurious correlations (see e.g. [ 16, 35 ]). We are rather going to define a particular class of distributions on the grand canonical phase space

D

^ε

:=

[

N≥0

D

^εN

,

where the particle number is not fixed but given by a modified Poisson law (actually D

^εN

= ∅ for large N ). For notational convenience, we work with functions extended to zero over

D^N

\ D

^εN

. Given a probability distribution f

⁰

:

D

→

R

satisfying

(1.1.5) | f

⁰

(x, v) | + |∇

^x

f

⁰

(x, v) | ≤ C

0

exp

− β

0

2 | v |

²

, C

0

> 0 , β

0

> 0 , the initial probability density is defined on the configurations (N, Z

N

) ∈

D^N

as

(1.1.6) 1

N! W

_N^ε0

(Z

N

) := 1 Z

^ε

µ

^N_ε

N !

N

Y

i=1

f

⁰

(z

i

) 1

D^ε_N

(Z

N

) where µ

ε

> 0 and the normalization constant Z

^ε

is given by

Z

^ε

:= 1 +

X

N≥1

µ

^N_ε

N !

Z

D^N

dZ

N N

Y

i=1

f

⁰

(z

i

) 1

_D^ε

N

(Z

N

) .

Here and below, 1

A

will be the characteristic function of the set A. We will also use the symbol 1

“∗”

for the characteristic function of the set defined by condition “ ∗ ”.

Note that in the chosen probability measure, particles are “exchangeable”, in the sense that W

_N^ε0

is invariant by permutation of the particle labels in its argument. Moreover, the choice (1.1.6) for the initial data is the one guaranteeing the “maximal factorization”, in the sense that particles would be i.i.d. were it not for the indicator function (‘hard-sphere exclusion’).

Our fundamental random variable is the time-zero configuration, consisting of the initial positions and velocities of all the particles of the gas. We will denote N the total number of particles (as a random variable) and Z

^ε0_N

= z

^ε0_i

i=1,...,N

the initial particle configuration. The particle dynamics (1.1.7) t 7→ Z

^ε_N

(t) = (z

^εi

(t))

_i=1,...,N

is then given by the hard-sphere flow solving (1.1.1)-(1.1.2) with random initial data Z

^ε0_N

(well defined

with probability 1). The probability of an event X with respect to the measure (1.1.6) will be de-

noted

Pε

(X ), and the corresponding expectation symbol will be denoted

Eε

. Notice that particles are

1.1. THE HARD-SPHERE MODEL 3

identified by their label, running from 1 to N . We shall mostly deal with expectations of observables of type

Eε PN

i=1

. . .

. Unless differently specified, we always imply that

Eε P

i

. . .

=

Eε PN i=1

. . .

. The average total number of particles N is fixed in such a way that

(1.1.8) lim

ε→0Eε

( N ) ε

^d−1

= 1 .

The limit (1.1.8) ensures that the Boltzmann-Grad scaling holds, i.e. that the inverse mean free path is of order 1 [19]. Thus from now on we will set

µ

ε

= ε

^−(d−1)

. Let us define the rescaled initial n-particle correlation function

F

_n^ε0

(Z

n

) := µ

⁻ⁿ_ε

∞

X

p=0

1 p!

Z

D^p

dz

n+1

. . . dz

n+p

W

_n+p^ε0

(Z

n+p

) .

We say that the initial measure admits correlation functions when the series in the right-hand side is convergent, together with the series in the inverse formula

W

_n^ε0

(Z

n

) = µ

ⁿ_ε

∞

X

p=0

( − µ

ε

)

^p

p!

Z

D^p

dz

n+1

. . . dz

n+p

F

_n+p^ε0

(Z

n+p

).

In this case, the set of functions F

_n^ε0

n≥1

describes all the properties of the system.

For any symmetric test function h

n

:

Dⁿ

→

R, the following holds :

(1.1.9)

Eε

X

i1,...,in

i_j6=ik,j6=k

h

n

z

^ε0_i1

, . . . , z

^ε0_in

=

Eε

δ

_{N ≥n}

N !

( N − n)! h

n

z

^ε0₁

, . . . , z

^ε0_n

=

∞

X

p=n

Z

D^p

dZ

_p

W

_p^ε0

(Z

p

) p!

p !

(p − n)! h

_n

Z

_n

= µ

ⁿ_ε Z

Dⁿ

dZ

_n

F

_n^ε0

( Z

_n

) h

_n

( Z

_n

) .

Starting from the initial distribution W

_N^ε0

, the density W

_N^ε

(t) evolves on D

^εN

according to the Liouville equation (1.1.3) with specular boundary reflection (1.1.4). At time t ≥ 0, the (rescaled) n-particle correlation function is defined as

F

_n^ε

(t, Z

n

) := µ

⁻ⁿ_ε

∞

X

p=0

1 p!

Z

D^p

dz

n+1

. . . dz

n+p

W

_n+p^ε

(t, Z

n+p

) (1.1.10)

and, as in (1.1.9), we get

(1.1.11)

Eε

X

i1,...,in

i_j6=ik,j6=k

h

n

z

^ε_i1

(t), . . . , z

^ε_in

(t)

= µ

ⁿ_ε Z

Dⁿ

dZ

n

F

_n^ε

(t, Z

n

) h

n

Z

n

,

where we used the notation (1.1.7). Notice that F

_n^ε

(t, Z

n

) = 0 for Z

n

∈

Dⁿ

\ D

n^ε

. In the following we shall denote the empirical measure

(1.1.12) π

^ε_t

:= 1

µ

ε N

X

i=1

δ

z^ε_i(t)

.

Tested on a (one-particle) function h :

D

→

R, it reads

(1.1.13) π

_t^ε

(h) = 1

µ

ε N

X

i=1

h (z

^εi

(t)) .

By definition, F

₁^ε

describes the average behavior of (exchangeable) particles :

(1.1.14)

E^ε

π

^ε_t

(h)

=

Z

D

F

₁^ε

(t, z) h(z) dz .

1.2. Lanford’s theorem : a law of large numbers

In the Boltzmann-Grad limit µ

ε

→ ∞ , the average behavior is governed by the Boltzmann equation : (1.2.1)







∂

_t

f + v · ∇

^x

f =

Z

R^d

Z

S^d−1

f (t, x, w

⁰

)f (t, x, v

⁰

) − f (t, x, w)f(t, x, v)

(v − w) · ω

+

dω dw , f (0, x, v) = f

⁰

(x, v)

where the precollisional velocities (v

⁰

, w

⁰

) are defined by the scattering law (1.2.2) v

⁰

:= v − (v − w) · ω

ω , w

⁰

:= w + (v − w) · ω ω .

More precisely, the convergence is described by Lanford’s theorem [ 28 ] (in the canonical setting — for the grand-canonical setting see [27], where the case of smooth compactly supported potentials is also addressed), which we state here in the case of the initial measure (1.1.6).

Theorem 1

(Lanford [28]). — Consider a system of hard spheres initially distributed according to the grand canonical measure (1.1.6) with f

⁰

satisfying the estimates (1.1.5). Then, in the Boltzmann- Grad limit µ

ε

→ ∞ , the rescaled one-particle density F

₁^ε

(t) converges uniformly on compact sets to the solution f (t) of the Boltzmann equation (1.2.1) on a time interval [0, T

0

] (which depends only on f

⁰

through C

₀

, β

₀

). Furthermore for each n, the rescaled n-particle correlation function F

_n^ε

(t) converges almost everywhere in

Dⁿ

to f

^⊗n

(t) on the same time interval.

We refer to [22, 44, 11, 17, 14, 6, 35] for details on this result and subsequent developments.

The propagation of chaos derived in Theorem 1 implies in particular that the empirical measure concentrates on the solution of Boltzmann equation. Indeed, computing the variance for any test function h, we get that

(1.2.3)

E^ε

π

^ε_t

(h) −

Z

F

₁^ε

(t, z) h(z) dz

²

=

Eε

1 µ

²_ε

N

X

i=1

h

²

z

^ε_i

(t) + 1

µ

²_ε X

i6=j

h z

^ε_i

(t)

h z

^ε_j

(t)

−

Z

F

₁^ε

(t, z) h(z) dz

2

= 1 µ

_ε

Z

F

₁^ε

(t, z) h

²

(z) dz +

Z

F

₂^ε

(t, Z

2

) h(z

1

)h(z

2

) dZ

2

−

Z

F

₁^ε

(t, z) h(z) dz

2

−−−−→

_µ

ε→∞

0 , where the convergence to 0 follows from the fact that F

₂^ε

converges to f

^⊗2

and F

₁^ε

to f almost everywhere. This computation can be interpreted as a law of large numbers and we have that, for all δ > 0, and smooth h,

(1.2.4)

Pε

π

_t^ε

(h) −

Z

D

f(t, z)h(z)dz

> δ

−−−−→

_µ

ε→∞

0 .

1.3. THE FLUCTUATING BOLTZMANN EQUATION 5

Remark 1.2.1. —

The restriction to the time interval [0, T

0

] in the statement of Theorem 1 is probably of technical nature: it originates from a Cauchy-Kowalevski argument in the Banach space of measur- able sequences F = (F

n

)

n≥1

with F

n

:

Dⁿ

→

R

, endowed with norm sup

_n≥1

sup

_Dn

| F

n

| e

^αn+β2|Vn|2

for suitable α, β ∈

R

.

1.3. The fluctuating Boltzmann equation

Describing the fluctuations around the Boltzmann equation is a way to capture part of the information which has been lost in the limit ε → 0.

As in the classical central limit theorem, we expect these fluctuations to be of order 1/ √ µ

_ε

, which is the typical size of the remaining correlations. We therefore define the fluctuation field ζ

^ε

as follows:

for any test function h :

D

→

R

(1.3.1) ζ

_t^ε

h

:= √ µ

_ε

π

_t^ε

(h) −

Z

F

₁^ε

(t, z) h z dz

.

Initially the empirical measure starts close to the density profile f

⁰

and ζ

₀^ε

converges in law towards a Gaussian white noise ζ

₀

with covariance

E

ζ

0

(h

1

) ζ

0

(h

2

)

=

Z

h

1

(z) h

2

(z) f

⁰

(z) dz .

In this paper we prove that in the limit µ

ε

→ ∞ , starting from “almost independent” hard spheres, ζ

^ε

converges to a Gaussian process, solving formally

(1.3.2) dζ

t

= L

^t

ζ

t

dt + dη

t

,

where L

^t

is the linearized Boltzmann operator around the solution f (t) of the Boltzmann equa- tion (1.2.1)

(1.3.3) L

^t

h(x, v) := − v · ∇

^x

h(x, v) +

Z

R^d

Z

S^d−1

dw dω (v − w) · ω

+

× f(t, x, w

⁰

)h(x, v

⁰

) + f (t, x, v

⁰

)h(x, w

⁰

) − f (t, x, v)h(x, w) − f (t, x, w)h(x, v) . The noise dη

t

(x, v) is Gaussian, with zero mean and covariance

(1.3.4)

E Z

dt

₁

dz

₁

h

₁

(z

1

)η

t1

(z

1

)

Z

dt

₂

dz

₂

h

₂

(z

2

)η

t2

(z

2

)

= 1 2

Z

dtdµ(z

1

, z

2

, ω)f (t, z

1

) f (t, z

2

)∆h

1

∆h

2

denoting

(1.3.5) dµ(z

1

, z

2

, ω) := δ

x₁−x2

(v

1

− v

2

) · ω

+

dω dv

1

dv

2

dx

1

and defining for any h

∆h(z

1

, z

₂

, ω) := h(z

₁⁰

) + h(z

₂⁰

) − h(z

1

) − h(z

2

) ,

where z

⁰_i

:= (x

i

, v

_i⁰

) with notation (1.2.2) for the velocities obtained after scattering. We postpone the precise definition of a weak solution to (1.3.2) to Section 6.1.2.

Our result is the following.

Theorem 2. —

Consider a system of hard spheres initially distributed according to the grand canon- ical measure (1.1.6) where f

⁰

is a smooth function satisfying (1.1.5). Then in the Boltzmann-Grad limit µ

ε

→ ∞ , the fluctuation field (ζ

_t^ε

)

_t≥0

converges in law to a Gaussian process solving (1.3.2) in a weak sense on a time interval [0, T

^?

].

The convergence towards the limiting process (1.3.2) was conjectured by Spohn in [43] and the non- equilibrium covariance of the process at two different times was computed in [42], see also [44]. The noise emerges after averaging the deterministic microscopic dynamics. It is white in time and space, but correlated in velocities so that momentum and energy are conserved.

At equilibrium the convergence of a discrete-velocity version of the same process at equilibrium was derived rigorously in [37], starting from a dynamics with stochastic collisions (see also [25, 24, 30]

for fluctuations in space-homogeneous models).

The physical aspects of the fluctuations for the rarefied gas have been thoroughly investigated in [16, 42, 43]. We also refer to [8], where we gave an outline of our results and strategy. Here we would like to recall only a few important features.

1) The noise in (1.3.2) originates from recollisions.

It is a very general fact that, when the macroscopic equation is dissipative, the dynamical equation for the fluctuations contains a term of noise. In the case under study, “recollisions” are a class of mechanical events giving a negligible contribution to the limit π

_t^ε

→ f (t) (see (1.2.4)) – for example, two particles colliding twice with each other in a finite time. The proof of Theorem 2 provides a further insight on the relation between collisions and noise. Following [42], we represent the dynamics in terms of a special class of trajectories, for which one can classify precisely the recollisions responsible for the term dη

t

; see Section 1.5 for further explanations. For the moment we just remind the reader that there is no a priori contradiction between the dynamics being deterministic, and the appearance of noise from collisions in the singular limit. Indeed when ε goes to zero, the deflection angles are no longer deterministic (as in the probabilistic interpretation of the Boltzmann equation). The randomness, which is entirely coded on the initial data of the hard sphere system, is transferred to the dynamics in the limit.

2) Equilibrium fluctuations can be deduced by the fluctuation-dissipation theorem.

As a particular case, we obtain the result at thermal equilibrium f

⁰

= M , where M is Maxwellian with inverse temperature β. The stochastic process (1.3.2) boils down to a generalized Ornstein- Uhlenbeck process. The noise term compensates the dissipation induced by the linearized Boltzmann operator, and the covariance of the noise (1.3.4) can be predicted heuristically by using the invariant measure [44].

3) Away from equilibrium, the fluctuating equations keep the same structure.

The most direct way to to guess (1.3.2)-(1.3.4) is starting from the equilibrium prediction (previous

point) and assuming that M = M (v) can be substituted with f = f (t, x, v). This heuristics is known

as “extended local equilibrium” assumption, in the context of fluctuating hydrodynamics. It is based

on the remark that the noise is white in space and time, and therefore only the local (in (x, t)) features

of the gas should be relevant. If the system has a “local equilibrium”, this is enough to determine the

equations. This procedure gives the right result also for our gas at low density (even if f = f (t, x, v)

is not locally Maxwellian). The reason is that a form of local equilibrium is still true; namely, around

1.4. LARGE DEVIATIONS 7

a little cube centered in x at time t, the hard sphere system is described by a Poisson measure with constant density

R

f (t, x, v)dv [44].

4) Away from equilibrium, fluctuations exhibit long range correlations.

The covariance of the fluctuation field (at equal times and) at different points x

₁

, x

₂

is not zero when | x

1

− x

2

| is of order one (and decays slowly with | x

1

− x

2

| ). This is typical of non equilibrium fluctuations [16]. In the hard sphere gas at low density, it is again related to recollisions, and the proof of Theorem 2 will provide an explicit formula quantifying this effect.

1.4. Large deviations

While typical fluctuations are of order O(µ

^−1/2ε

), they may sometimes happen to be large, leading to a dynamics which is different from the Boltzmann equation. A classical problem is to evaluate the probability of such an atypical event, namely that the empirical measure remains close to a probability density ϕ 6 = f during a time interval [0 , T

^?

]. The following explicit formula for the large deviation functional was obtained by Rezakhanlou [38] in the case of a one-dimensional stochastic dynamics mimicking the hard-sphere dynamics, and then conjectured for the three-dimensional deterministic hard-sphere dynamics by Bouchet [9]:

F

b

(t, ϕ) := F

b

(0, ϕ

0

) + sup

p

Z t 0

ds

Z

T^d

dx

Z

R^d

dv p(s, x, v) D

_s

ϕ(s, x, v) − H ϕ(s), p(s)

(1.4.1) ,

where the supremum is taken over bounded measurable functions p, and the Hamiltonian is given by

(1.4.2) H (ϕ, p) := 1

2

Z

dµ(z

1

, z

2

, ω)ϕ(z

1

)ϕ(z

2

) exp ∆p

− 1 . We have denoted D

t

the transport operator

(1.4.3) D

t

ϕ(t, z) := ∂

t

ϕ(t, z) + v · ∇

^x

ϕ(t, z) , and finally

(1.4.4) F

b

(0, ϕ

0

) :=

Z

D

dz

ϕ

₀

log ϕ

₀

f

⁰

− ϕ

₀

+ f

⁰

with ϕ

0

= ϕ |

^t=0

, is the large deviation rate for the empirical measure at time zero. F

b

(0) can be obtained by a standard procedure, modifying the measure (1.1.6) in such a way to make the (atypical) profile ϕ

₀

typical. Similarly, to obtain the collisional term H in F

b

( t, ϕ ), one would like to understand the mechanism leading to an atypical path ϕ = ϕ(t) at positive times. A serious difficulty then arises, due to the deterministic dynamics. Ideally, one should conceive a way of tilting the initial measure in order to observe a given trajectory. Whether such an efficient bias exists, we do not know. But we shall proceed in a different way, inspecting somehow the dynamics at all scales in ε. This strategy, which will be informally described in the next section, leads to Theorem 3. The remarkable feature of this result is that the large deviation behaviour of the mechanical dynamics is also ruled by the large deviation functional of the stochastic process.

Denote by M the set of probability measures on

D

(with the topology of weak convergence) and

by D([0, T

^?

], M ) the Skorokhod space (see [4] page 121).

Theorem 3. —

Consider a system of hard spheres initially distributed according to the grand canonical measure (1.1.6) where f

⁰

satisfies (1.1.5). There exists a time T

^?

and a functional F = F (T

^?

, · ) such that, in the Boltzmann-Grad limit µ

ε

→ ∞ , the empirical measure satisfies the following large deviation estimates :

— For any compact set F ⊂ D([0, T

^?

], M ), lim sup

µ_ε→∞

1 µ

ε

log

Pε

(π

^ε

∈ F) ≤ − inf

ϕ∈F

F (T

^?

, ϕ) ; (1.4.5)

— For any open set O ⊂ D ([0 , T

^?

] , M ) , lim inf

µε→∞

1 µ

ε

log

Pε

(π

^ε

∈ O) ≥ − inf

ϕ∈O∩R

F (T

^?

, ϕ) , (1.4.6)

where R is a non trivial subset of D ([0 , T

^?

] , M ) .

Moreover there exists a non trivial subset of R , and a time T ≤ T

^?

, such that the functionals F (T, · ) and F ˆ (T, · ) coincide on R .

The functional F is determined by the solution of a variational problem (see (7.0.3) below) and the set R is chosen such that the extremum of this variational principle is attained in a class of sufficiently small and regular functions: see (7.1.5).

For an extensive formal discussion on large deviations in the Boltzmann gas, we refer to [9].

1.5. Strategy of the proofs

In this section we provide an overview of the paper and describe, informally, the core of our argument leading to Theorems 2 and 3.

We should start recalling the basic features of the proof of Theorem 1. For a deterministic dynamics of interacting particles, so far there has been only one way to access the law of large numbers rigor- ously. The strategy is based on the ‘hierarchy of moments’ corresponding to the family of correlation functions (F

_n^ε

)

_n≥1

, Eq. (1.1.10). The main role of F

_n^ε

is to project the measure on finite groups of particles (groups of cardinality n), out of the total N . The term ‘hierarchy’ refers to the set of linear BBGKY equations satisfied by this collection of functions (which will be written in Section 3.1), where the equation for F

_n^ε

has a source term depending on F

_n+1^ε

. This hierarchy is completely equivalent to the Liouville equation (1.1.3) for the family (W

_N^ε

)

_N≥0

, as it contains exactly the same amount of information. However as N ∼ µ

ε

in the Boltzmann-Grad limit (1.1.8), one should make sense of a Liouville density depending on infinitely many variables, and the BBGKY hierarchy becomes the natural convenient way to grasp the relevant information. Lanford succeeded to show that the ex- plicit solution F

_n^ε

(t) of the BBGKY, obtained by iteration of the Duhamel formula, converges to a product f

^⊗n

(t) (propagation of chaos), where f is the solution of the Boltzmann equation (1.2.1).

The hierarchy of moments has two important limitations. The first one is the restriction on its time of

validity, which comes from too many terms in the iteration: we are indeed unable to take advantage

of cancellations between gain and loss terms. The second one is a drastic loss of information. We shall

not give here a precise notion of ‘information’. We limit ourselves to stressing that (F

_n^ε

)

_n≥1

is suited

to the description of typical events. In the limit, everything is encoded in f , no matter how large n.

1.5. STRATEGY OF THE PROOFS 9

Moreover, the Boltzmann equation produces some entropy along the dynamics: at least formally, f satisfies

∂

t

−

Z

f log f dv

+ ∇

^x

· −

Z

f log f v dv

≥ 0 ,

which is in contrast with the time-reversible hard-sphere dynamics. Our main purpose here is to overcome this second limitation (for short times) and to perform the Boltzmann-Grad limit in such a way as to keep most of the information lost in Theorem 1. In particular, the limiting functional (1.4.1) coincides with the large deviations functional of a genuine reversible Markov process, in agreement with the microscopic reversibility [9]. We face a significant difficulty: on the one hand, we know that averaging is important in order to go from Newton’s equation to Boltzmann’s equation; on the other hand, we want to keep track of some of the microscopic structure.

To this end, we need to go beyond the BBGKY hierarchy and turn to a more powerful representation of the dynamics. We shall replace the family (F

_n^ε

)

_n≥1

(or (W

_N^ε

)

_N≥0

) with a third, equivalent, family of functions ( f

_n^ε

)

_n≥1

, called (rescaled) cumulants

⁽¹⁾

. Their role is to grasp information on the dynamics on finer and finer scales. Loosely speaking, f

_n^ε

will collect events where n particles are “completely connected” by a chain of interactions. We shall say that the n particles form a cluster. Since a collision between two given particles is typically of order µ

⁻¹_ε

, a “complete connection” would account for events of probability of order µ

⁻⁽ⁿ⁻¹⁾ε

. We therefore end up with a hierarchy of rare events, which we need to control at all orders to obtain Theorem 3. At variance with (F

_n^ε

)

_n≥1

, even after the limit µ

ε

→ ∞ is taken, the rescaled cumulant f

_n^ε

cannot be trivially obtained from the cumulant f

_n−1^ε

. Each step entails extra information, and events of increasing complexity, and decreasing probability.

The cumulants, which are a standard probabilistic tool, will be investigated here in the dynamical, non-equilibrium context. Their precise definition and basic properties are discussed in Chapter 2.

The introduction of cumulants will not entitle us to avoid the BBGKY hierarchy entirely. Un- fortunately, the equations for (f

_n^ε

)

_n≥1

are difficult to handle. But the moment-to-cumulant rela- tion (F

n^ε

)

_n≥1

→ (f

n^ε

)

_n≥1

is a bijection and, in order to construct f

_n^ε

(t), we can still resort to the same solution representation of [28] for the correlation functions (F

_n^ε

(t))

_n≥1

. This formula is an expansion over collision trees, meaning that it has a geometrical representation as a sum over binary tree graphs, with vertices accounting for collisions. The formula will be presented in Chapter 3 (and generalized from the finite-dimensional case to the case of functionals over trajectories, which is needed to deal with space-time processes). For the moment, let us give an idea of the structure of this tree expansion.

The Duhamel iterated solution for F

_n^ε

(t) has a peculiar characteristic flow: n hard spheres (of diame- ter ε) at time t flow backwards, and collide (among themselves or) with a certain number of external particles, which are added at random times and at random collision configurations. The following picture is an example of such flow (say, n = 3):

.

1. Cumulant type expansions within the framework of kinetic theory appear in [5, 35, 29, 18]

The net effect resembles a binary tree graph. The real graph is just a way to record which pairs of particles collided, and in which order.

It is important to notice that different subtrees are unlikely to interact: since the hard spheres are small and the trajectories involve finitely many particles, two subtrees will encounter each other with small probability. This is a rather pragmatic point of view on the propagation of chaos, and the reason why F

_n^ε

(t) is close to a tensor product (if it is so at time zero) in the classical Lanford argument.

Observe that, in this simple argument, we are giving a notion of dynamical correlation which is purely geometrical. Actually we will use this idea over and over. Two particles are correlated if their generated subtrees are connected , as represented for instance in the following picture:

which is an event of ‘size’ µ

⁻¹_ε

(the volume of a tube of diameter ε and length 1). In Chapter 4, we will give precise definitions of correlation (connection) based on geometrical constraints. It will be the elementary brick to characterize f

_n^ε

(t) explicitly in terms of the initial data. The formula for f

_n^ε

(t) (Section 4.4) will be supported on characteristic flows with n particles connected, through their generated subtrees (hence of expected size µ

⁻⁽ⁿ⁻¹⁾ε

). In other words, while F

_n^ε

projects the measure on arbitrary groups of particles of size n, the improvement of f

_n^ε

consists in restricting to completely connected clusters of the same size.

With this naive picture in mind, let us briefly comment again on information, and irreversibility. One nice feature of the geometric analysis of recollisions is that it reflects the transition from a time- reversible to a time-irreversible model. In [7] we identified, and quantified, the microscopic singular sets where F

_n^ε

does not converge. These sets are not invariant by time-reversal (they have a direction always pointing to the past, and not to the future). Looking at F

_n^ε

(t), we lose track of what happens in these small sets. This implies, in particular, that Theorem 1 cannot be used to come back from time t > 0 to the initial state at time zero. The cumulants describe what happens on all the small singular sets, therefore providing the information missing to recover the reversibility.

At the end of Chapter 4, we give a uniform estimate on these cumulants (Theorem 4), which is the

main advance of this paper. This L

¹

-bound is sharp in ε and n ( n -factorial bound), roughly stating

that the unscaled cumulant decays as µ

⁻⁽ⁿ⁻¹⁾ε

n

ⁿ⁻²

. This estimate is intuitively simple. We have given

a geometric notion of correlation as a link between two collision trees. Based on this notion, we can

draw a random graph telling us which particles are correlated and which particles are not (each collision

tree being one vertex of the graph). Since the cumulant describes n completely correlated particles,

there will be at least n − 1 edges, each one of small ‘volume’ µ

⁻¹_ε

. Of course there may be more than

n − 1 connections (the random graph has cycles), but these are hopefully unlikely as they produce

extra smallness in ε. If we ignore all of them, we are left with minimally connected graphs, whose total

number is n

ⁿ⁻²

by Cayley’s formula. Thanks to the good dependence in n of these uniform bounds,

we can actually sum up all the family of cumulants into an analytic series, referred to as ‘cumulant

generating function’.

1.6. REMARKS, AND OPEN PROBLEMS 11

The second central result of this paper, stated in Chapter 5 (Theorem 5), is the characterization of the rescaled cumulants in the Boltzmann-Grad limit, with minimally connected graphs. Using this minimality property, we actually derive a Hamilton-Jacobi equation for the limiting cumulant generating function. Wellposedness and uniqueness for this equation can be achieved by abstract methods, based on analyticity. All the information of the microscopic mechanical model is actually encoded in this Hamilton-Jacobi equation which, in particular, allows us to characterize the large deviation functional. which is our ultimate point of arrival. From this Hamilton-Jacobi equation, we can also obtain differential equations for the limiting family of cumulants (f

n

)

_n≥1

. These equations, which we may call Boltzmann cumulant hierarchy, have a remarkable structure and have been written first in [16].

The rest of the paper is devoted to the proofs of our main results.

Chapter 6 proves Theorem 2. Here, the uniform bounds of Theorem 4 are considerably better than what is required, and the proof amounts to looking at a characteristic function living on larger scales.

The more technical part of the proof concerns the tightness of the process for which we adapt a Garsia-Rodemich-Rumsey’s inequality on the modulus of continuity, to the case of a discontinuous process.

In Chapter 7 we prove Theorem 3. Our purpose is to show that the functional obtained in Chapter 5 is dual, through the Legendre transform, to a large deviation rate function. In the absence of global convexity, we will not succeed in proving a full large deviation principle. However, restricting to a class of regular profiles, the variational problem is uniquely solved and the rate functional can be identified with the one predicted in the physical literature, based on the analogy with stochastic dynamics.

Finally, Chapters 8 and 9 are devoted to the proof of Theorems 4 and 5, respectively. We encounter here a combinatorial issue. The number of terms in the formula for f

_n^ε

( t ) grows, at first sight, badly with n, and cancellations need to be exploited to obtain a factorial growth. At this point, cluster expansion methods ([39]) enter the game (summarized in Chapter 2), applied to the collision trees.

The decay µ

⁻⁽ⁿ⁻¹⁾ε

follows instead from a geometric analysis on hard-sphere trajectories with n − 1 connecting constraints, in the spirit of previous work [ 5, 7, 35 ].

1.6. Remarks, and open problems

We conclude with a few remarks on our results.

— To simplify our proof, we assumed that the initial datum is a quasi-product measure, with the minimal amount of correlations (only the mutual exclusion between hard spheres is taken into account). This assumption is useful to isolate the dynamical part of the problem in the clearest way. More general initial states could be dealt with along the same lines ([ 43, 35 ]). However the cumulant expansions would contain more terms, describing the deterministic (linearized) transport of initial correlations.

— Similarly, fixing only the average number of particles (instead of the exact number of particles)

allows to avoid spurious correlations. We therefore work in a grand canonical setting, as is

customary in statistical physics when dealing with fluctuations. Notice that fixing N = N

produces a long range term of order 1/N in the covariance of the fluctuation field. Note also

that the cluster expansion method, which is crucial in our analysis, is developed (with few exceptions, see [36] for instance) in a grand canonical framework [33].

— Our results could be established in the whole space

R^d

, or in a parallelepiped box with periodic or reflecting boundary conditions. Different domains might be also covered, at the expense of complications in the geometrical estimates of recollisions (see [ 15 ] for instance).

— We do not deal with the original BBGKY hierarchy of equations, which was written for smooth potentials, but always restrict to the hard-sphere system. It is plausible that our results could be extended to smooth, compactly supported potentials as considered in [17, 34] (see [2] for a fast decaying case), but the proof would be considerably more involved.

— At thermal equilibrium, we expect Theorem 2 to be true globally in time: see [5] for a first step

in this direction.

PART I

DYNAMICAL CUMULANTS

CHAPTER 2

COMBINATORICS ON CONNECTED CLUSTERS

This preliminary chapter consists in presenting a few notions (well-known in statistical mechanics) that will be essential in our analysis. We present in particular cumulants, and their link with expo- nential moments as well as with cluster expansions. We conclude the chapter with some combinatorial identities that will be useful throughout this work.

2.1. Generating functionals and cumulants

Let h :

D

→

R

be a bounded continuous function. We shall use the notation

(2.1.1)

F

_n^ε

(t), h

^⊗n

=

Z

Dⁿ

dZ

n

F

_n^ε

(t, Z

n

)h(z

1

) . . . h(z

n

) , and

P

n^s

= set of partitions of { 1, . . . , n } into s parts , with

σ ∈ P

n^s

= ⇒ σ = { σ

1

, . . . , σ

s

} , | σ

i

| = κ

i

,

s

X

i=1

κ

i

= n .

The moment generating functional of the empirical measure (1.1.13), namely

Eε

exp π

^ε_t

(h) is re- lated to the rescaled correlation functions (1.1.10) by the following remark. We recall that

(2.1.2)

Eε

exp π

^ε_t

(h)

=

Eε

"

exp 1 µ

ε

N

X

i=1

h z

^ε_i

(t)

#

.

Proposition 2.1.1. —

We have that

(2.1.3)

Eε

exp π

^ε_t

(h)

= 1 +

∞

X

n=1

µ

ⁿ_ε

n!

F

_n^ε

(t),

e

^h/µε

− 1

⊗n

if the series is absolutely convergent.

Proof. — Starting from (2.1.2), one has

X

k≥1

1 k!

Eε

π

_t^ε

(h)

k

=

X

k≥1

1 k!

k

X

n=1

X

σ∈P_kⁿ

µ

^−k_ε Eε

X

i₁,...,i_n ij6=i<sub>`</sub>,j6=`

h z

^ε_i

1

(t)

^κ1

. . . h z

^ε_i

n

(t)

^κn

=

X

k≥1

1 k!

k

X

n=1

X

σ∈P_kⁿ

µ

^−k_ε

µ

ⁿ_ε Z

Dⁿ

dZ

n

F

_n^ε

(t, Z

n

)h(z

1

)

^κ1

. . . h(z

n

)

^κn

where in the last equality we used (1.1.11). On the other hand for fixed n

X

k≥n

µ

^−k_ε

k!

X

σ∈P_kⁿ n

Y

i=1

h ( z

_i

)

^κi

=

X

k≥n

µ

^−k_ε

k! n!

X

κ₁···κn≥1 Pκi=k

k κ

1

k − κ

₁

κ

2

· · ·

k − κ

₁

− · · · − κ

_n−2

κ

_n−1

ⁿ Y

i=1

h ( z

_i

)

^κi

= 1 n!

n

Y

i=1

X

κ_i≥1

h(z

i

)

^κi

µ

^κεⁱ

κ

i

! = 1

n!

n

Y

i=1

e

^h(zi)/µε

− 1 . Therefore

Eε

exp

π

_t^ε

(h)

= 1 +

X

n≥1

µ

ⁿ_ε Z

Dⁿ

dZ

n

F

_n^ε

(t, Z

n

) 1 n!

n

Y

i=1

e

^h(zi)/µε

− 1 , which proves the proposition.

The moment generating functional is just a compact representation of the information coded in the family ( F

_n^ε

( t ))

_n≥1

. After the Boltzmann-Grad limit µ

_ε

→ ∞ , the right-hand side of (2.1.3) reduces to

∞

X

n=0

1 n!

Z

f (t)h

n

= exp

Z

f (t)h

, i.e. to the solution of the Boltzmann equation.

As discussed in the introduction, our purpose is to keep a much larger amount of information. To this end, we study the cumulant generating functional which is, by Cram´er’s theorem, an obvious candidate to reach atypical profiles [46]. Namely, we pass to the logarithm and rescale as follows:

(2.1.4) Λ

^εt

( e

^h

) := 1 µ

ε

log

Eε

exp

µ

_ε

π

^ε_t

( h )

= 1 µ

ε

log

Eε

exp

N

X

i=1

h z

^ε_i

( t ) .

The first task is to look for a proposition analogous to the previous one. In doing so, the following definition emerges naturally, where we use the notation:

(2.1.5) G

_σj

:= G

_|σj|

( Z

_σj

) , G

_σ

:=

|σ|

Y

j=1

G

_σj

for σ = { σ

₁

, . . . , σ

_s

} ∈ P

n^s

.

Definition 2.1.2

(Cumulants). — Let (G

n

)

n≥1

be a family of distributions of n variables invari- ant by permutation of the labels of the variables. The cumulants associated with ( G

_n

)

n≥1

form the family (g

n

)

n≥1

defined, for all n ≥ 1, by

g

n

:=

n

X

s=1

X

σ∈P^s_n

( − 1)

^s−1

(s − 1)! G

σ

.

We then have the following result, which is well-known in the theory of point processes (see [12]).